This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the first author is also supported by grant Fundaci´o Bancaixa P11B2010-47. The second author would like to thank the members of the Institute of Mathematics at the University of Aberdeen for their warm hospitality and for many helpful discussions during her visit to this University.

[EN] We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes. We show that if the set of G-class sizes of N is {1, m, mpa }, with p a prime not dividing m, then N is solvable. Thus, we get a partial extension for normal subgroups on N. Itˆo’s theorem on the solvability of groups having exactly three class sizes.